Welcome to the official WWW

ALPH
.A.
-----
METIC

Page

Yes, this is it! The official one and only Alphametic Page,
dedicated to that most elegant of puzzles (combining
mathematical and word play) with which I have been obsessed,
on and off, for my entire adult life. If you've never seen an
alphametic, I'll show you what the fuss is all about. If you have,
I will try to regale you some of my own creations that have
pushed the envelope of alphametic possibilities to new and
bizarre heights.

Some History and Philosophy

An alphametic is a peculiar type of
mathematical puzzle, in which a set of words is written down
in the form of an ordinary "long-hand" addition
sum, and it is required that the letters of the alphabet be
replaced with decimal digits so that the result is a valid
arithmetic sum. For an example one can do no better than the
first modern alphametic, published by the great puzzlist H.E.
Dudeney in the July 1924 issue of Strand Magazine:

SEND
MORE
-----
MONEY

whose (unique) solution is:

9567
1085
-----
10652

There are two fairly obvious
(but worth stating) rules which every alphametic obeys:

1. The mapping of letters to numbers is
one-to-one. That is, the same letter always stands for
the same digit, and the same digit is always represented
by the same letter.

2. The digit zero is not allowed to
appear as the left-most digit in any of the addends or
the sum.

Why are alphametics so cool? For one thing,
there is their economy: with only a few words, a puzzle that
can easily take half an hour to solve can be written down.
The process of solving an alphametic is itself interesting,
often illustrating the triumph of logic over trial and error.
The puzzle above (SEND + MORE = MONEY) is especially elegant
in this regard - it can be solved in a matter of seconds via
a few observations.

The other obvious attraction is that
alphametics are hard to construct. First of all, since we
usually deal in base 10, only 10 different letters of the
alphabet (at most) can be used. This, naturally, makes it
hard to write phrases or sentences that read well. (There is
a vague analogy here to the difficulty of writing a long
palindrome that reads well.) Even if we write down a nice
phrase or sentence representing a prospective alphametic, the
odds that the alphametic will actually be solvable are pretty
small. Finally, there is what I consider the most important
feature an alphametic should have (but which imposes an
additional harsh constraint on the constructor):

Any truly elegant alphametic should
have a unique solution.

This condition of uniqueness is often not
required by alphametic constructors. They hack around this
difficulty by presenting the problem with a "side
condition", such as "make the sum a prime
number" or some such statement. In my opinion this is
very inelegant, and so on this page I only allow
alphametics that have a unique solution.

The problem of alphametic construction can
essentially be thought of as a very difficult form of constrained
writing: the object is to write a phrase or sentence that
(a) reads well, and (b) when considered as an alphametic
(with the last word being the sum word), it is solvable
(preferably with a unique solution). For the last ten years
or so, I have primarily concentrated on this
"meta-problem" - the construction of
ever-more-elaborate alphametics that have a unique solution.

In the following sections, I'll present the
creme de la creme of alphametic puzzledom - both
traditional alphametics devised by others as well as some of
my unusual creations. Every alphametic presented here has a
unique solution, which you are hereby encouraged to find!

By the way, the title of this page, ALPH
+ .A. + METIC is a uniquely-solvable alphametic, where
the period is considered one of the ten symbols to be
replaced with digits. In fact, in 1995 I proved that this
alphametic is the smallest uniquely-solvable alphametic
containing all 10 digits. It is rather elegant that the
smallest such alphametic contains the word
"alphametic", is it not?

In this section I list some of the nicer
"traditional" alphametics I've seen over the years.
For conciseness, I just write them as sentences. It is
understood that the first n-1 words are the addends
and the nth word is the sum, that case is to be
ignored, and that punctuation is not part of the alphametic
(only the words). There are basically two types: phrases or
lists, and complete sentences.

The last one here is a standout, because it
reads well, is fairly long, and contains some long words. It
also does a very nice job of not looking like it only
contains 10 distinct letters. This alphametic inspired me to
tackle the following interesting question: what's the longest
word that can be worked into a uniquely-solvable alphametic?
My best effort can be found in the New Literary Frontiers section.

The doubly-true alphametic is an important
sub-genre of alphametic puzzledom that made its first
appearance in 1969. Since then, the creator of this first
puzzle, Steven Kahan, has published literally hundreds more
and has elevated the doubly-true alphametic to a high art
form.

A doubly-true alphametic is
one with the following remarkable property: the addends and
the sum are "number words", and when read as words
they also form a valid addition sum. Here is a simple
example:

THREE
THREE
TWO
TWO
ONE
------
ELEVEN

This, as a matter of fact,
is the "smallest" doubly-true English alphametic
with unique solution, where "smallest" means having
the smallest sum word (11). In 1994 I conducted an exhaustive
search of the approximately one million doubly-true
alphametics with sum word less than FIFTY, and found that
there are exactly 266 with a unique solution. Here are some
more examples:

In the "Traditional" section I
present some examples of nice phrases or sentences that are
uniquely-solvable alphametics. There's a vast, relatively
unexplored, expanse beyond the humble sentence: longer
sentences, poetry, special kinds of sentences (e.g.,
palindromes), and so on. It's not inconcievable that one
could write an entire story in which each sentence is a
uniquely-solvable alphametic. In this section I present some
alphametics I've constructed that open up new territory in
the literary realm.

First, a few poems. Here is a native
nursery rhyme from the mythical island of Sevvoth that lies
in the midst of the North Sea. In this poem, the words in the
poem are the addends and the title of the poem is the sum
word.

Sevvoth

Ten herons rest near North Sea shore
As tan terns soar to enter there.
As herons nest on stones at shore,
Three stars are seen; tern snores are near!

Note that this poem has perfect meter. A
different type of constraint is that imposed by the haiku,
which consists of exactly three lines of 5, 7, and 5
syllables. Here is a haiku (again, with the title being the
sum word) inspired by contemplating the flatness of glacial
ice sheets:

Flatiana

In Arctic terrain
An ancient, eerie ice tract
I enter a trance

Here are two examples of slightly longer
narratives (beyond the single sentence). In both cases, the
last word in the narrative is the sum word.

A tree is a rare treat. I siesta as
I sit at it. I stir as I stare at sea. As I sit, stars
rise nicely.

Dad and son sat on sod. Dad stood at
noon, and son soon stood, too (on sand, not sod). A sad
son, sans Dad, stood and stood. No Dad. A sad, sad
finale.

Finally, here is my best attempt so far to
incorporate long words into an alphametic. This alphametic -
a free-verse poem of sorts - contains a 17-letter word, which
I believe is the current world record, along with a large
number of other long words. Once again, the last word is the
sum.

In closing, here's an unsolved challenge:
find a sentence that is a uniquely-solvable alphametic and
also a palindrome. So far I have found several examples that
are near-misses, having exactly two solutions (instead of the
desired one). Here is one, a hypothetical headline from the Weekly
World News for July 20, 1969:

which is nearly a quote from Act IV, Scene
1 of Macbeth, missing only the 'and' before 'trouble'.
Many years later, this inspired the search for
"found" alphametics - exact quotes from literature
that just happen to be uniquely-solvable alphametics. Here
are a few I've found:

Lest you think that alphametic
possibilities are limited to words, here is a rather unusual
type of alphametic I invented about twenty years ago - the
chess alphametic, or chessametic.

A good representative chessametic is the
first one I published, in Vol. 8 No. 4 of the Journal of
Recreational Mathematics (1975). Starting from the
initial position in chess, consider the following legal chess
game:

Amazingly, if we write all the moves of the
game in a single column, with White's final exclamation at
the bottom, we obtain a solvable alphametic, where the ten
symbols to be replaced by digits are P,K,B,R,Q,I,W,N,-, and
x:

P-K4
P-K4
B-B4
P-R4
Q-B3
P-R4
QxP
----
IWIN

In a chessametic (or indeed, in any
alphametic where some digits are already given), any digits
already shown in the puzzle are simply to be left as is, and
are still available to be substituted for one of the symbols.

This puzzle has two slight shortcomings
from a chess point of view: (1) the two P-R4 moves are
ambiguous (they could mean either KR4 followed by QR4 or the
reverse), but this makes no difference in the final game
position, and (2) in the last move (QxP), the fact that it is
Black's KBP that is captured has to be inferred from the
stated fact that this move is checkmate. These flaws are
somewhat offset by the fact that this chessametic is of a
very special kind, since it starts with the chess pieces in
their initial position. It is, of course, much easier to
construct a chess alphametic where the starting position as
well as the moves are specified (as opposed to just the
moves).

Here is another chessametic for your
amusement. This one is much longer (12½ moves!), also
begins from the starting position, has no ambiguities in the
chess notation, is a much nicer chess game, and as an
alphametic has a unique solution. In short, it's perfect!

Many ideas come to mind for other
variations on this theme. I have seen at least one published
chessametic where the game ends in a draw and the sum word is
DRAW. It would be nice to have a classical chess problem in
which, following the key move, there are several lines of
play depending on Black's reply, each of which forms a
uniquely-solvable alphametic. Finally, there are many other
games (e.g., checkers) from which the notation for a set of
moves could be made into an alphametic. Some day, when I have
nothing better to do...

This nxn array has an amazing
property. If we attempt to solve it as an alphametic (with
the bottom word being the sum), subject to the side condition
of trying to maximize the numerical value of the sum, it has
a unique solution. The same alphametic with the side
condition "make the value of the sum minimum" also
has a unique solution. If we rotate the array by 90 degrees,
the resulting array also has a unique solution under
each of the two side conditions! And so does the array
rotated by 180 and 270 degrees. So, in all, there are eight
interesting puzzles contained within this single array.

This is a 4x4 alphametic word square. Do
larger word squares exist (e.g., 5x5)? Is it possible to make
one that has words reading both horizontally and vertically
(i.e., a traditional word square)? Is it possible to make one
that has a single unique solution for all four puzzles
(instead, as in this one, a unique solution for the
conditions max and min sum)? These are all unsolved problems.

For an even more elaborate alphametic word
square, see my puzzle L=-U .

Here is a very bizarre type of alphametic:
the order-n alphametic. A simple example is

Thy hay myth.

where, as usual the last word is the sum.
The goal of this puzzle is multifaceted. First, find its
unique solution. Then, take each of the three numbers (the
two addends and the sum) in the solution and replace each
number (m) with f(m) = the last M
digits of m² (where M is the number of
digits of m). For example, if m=234, you would
square it to get 54756, then take the last three digits
(756). Next, replace all the digits in the result by letters
using the rule (A=1, B=2, ... J=10). Consider what you have
now as an alphametic, and solve it again! Then apply the
function f(n) again, convert to letters, and
solve again! In this remarkable puzzle, all three alphametics
have a unique solution.

An alphametic of this type, in which n
uniquely-solvable alphametics are cleverly concealed, is
referred to as an order-n alphametic (under some
function f(m), which must be specified).

I have constructed one order-4 alphametic,
which you are hereby challenged to solve:

Tar star tree.

(with the same function f(m)
as the one above).

So far I haven't devised any with order
greater than 4, or with more than three words. But I'm sure
they're out there waiting to be discovered...